One blower I have a line on has a square chute, but it also has an enclosed reel. I think someone fab'd up the shroud. It looks to be in good condition, but it's about 3 hours from here so dropping by to see it isn't a real option at the moment. The other one I haven't heard back as to what model it is, and it's 4 hours away.
I have seen a few of the older square shooters that were enclosed, many of them were postal units by the sellers descriptions.
I use a square shooter,, never had a newer model, but it works ok. Some of what I see as disadvantages are the inability to direct the snow discharge in any direction except to the right or left and not having that hand crank to adjust the direction. The open reel is of course something to be careful of, not only for yourself and the pets, but any nearby object,,, I bit a car bumper with mine once trying to get to close, it was very unforgiving even though the slip clutch went off. The impeller on the very early ones just used flat plate fins and are easily damaged by stones, the later impeller is tougher in my opinion but both suffer from just enough clearance between the impeller and housing that small stones wedge in between and of course jam the impeller, probably the reason they dont toss the snow as far,, not a prob if your using it on pavement, but on gravel it can be.
What I like about the square shoot,,,, its simple, not much rocket science whatsoever,, and my thoughts are even if you cant find some parts it would be very easy to make or adapt them.